Theorem ineccnvmo 38339

Description: Equivalence of a double restricted universal quantification and a restricted "at most one" inside a universal quantification. (Contributed by Peter Mazsa, 2-Sep-2021.)

Assertion

Ref	Expression
ineccnvmo	⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑦 = 𝑧 ∨ ([𝑦]◡𝐹 ∩ [𝑧]◡𝐹) = ∅) ↔ ∀𝑥∃*𝑦 ∈ 𝐵 𝑥𝐹𝑦)

Distinct variable groups:   𝑥,𝐵,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧

Proof of Theorem ineccnvmo

Step	Hyp	Ref	Expression
1		relcnv 6125	. . 3 ⊢ Rel ◡𝐹
2		id 22	. . . 4 ⊢ (𝑦 = 𝑧 → 𝑦 = 𝑧)
3	2	inecmo 38337	. . 3 ⊢ (Rel ◡𝐹 → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑦 = 𝑧 ∨ ([𝑦]◡𝐹 ∩ [𝑧]◡𝐹) = ∅) ↔ ∀𝑥∃*𝑦 ∈ 𝐵 𝑦◡𝐹𝑥))
4	1, 3	ax-mp 5	. 2 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑦 = 𝑧 ∨ ([𝑦]◡𝐹 ∩ [𝑧]◡𝐹) = ∅) ↔ ∀𝑥∃*𝑦 ∈ 𝐵 𝑦◡𝐹𝑥)
5		brcnvg 5893	. . . . 5 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦◡𝐹𝑥 ↔ 𝑥𝐹𝑦))
6	5	el2v 3485	. . . 4 ⊢ (𝑦◡𝐹𝑥 ↔ 𝑥𝐹𝑦)
7	6	rmobii 3386	. . 3 ⊢ (∃*𝑦 ∈ 𝐵 𝑦◡𝐹𝑥 ↔ ∃*𝑦 ∈ 𝐵 𝑥𝐹𝑦)
8	7	albii 1816	. 2 ⊢ (∀𝑥∃*𝑦 ∈ 𝐵 𝑦◡𝐹𝑥 ↔ ∀𝑥∃*𝑦 ∈ 𝐵 𝑥𝐹𝑦)
9	4, 8	bitri 275	1 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑦 = 𝑧 ∨ ([𝑦]◡𝐹 ∩ [𝑧]◡𝐹) = ∅) ↔ ∀𝑥∃*𝑦 ∈ 𝐵 𝑥𝐹𝑦)

Syntax hints:   ↔ wb 206   ∨ wo 847  ∀wal 1535   = wceq 1537  ∀wral 3059  ∃*wrmo 3377  Vcvv 3478   ∩ cin 3962  ∅c0 4339   class class class wbr 5148  ^◡ccnv 5688  Rel wrel 5694  [cec 8742

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rmo 3378  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ec 8746

This theorem is referenced by:  ineccnvmo2  38342